## 1. Introduction

[2] It is well known that spatial and/or temporal functions representing the structure of the Earth constrained by finite geophysical observations must be square integrable. That is, observations manifested from Earth structures according to well-established, deterministic physical laws provide finite bandwidth information concerning the Earth model. In other words, a fair approximation of the Earth can only be constituted in the finite degrees of freedom (d.o.f.) provided by the finite amount of noisy and band-limited data constraints. However, it is not possible to make an adequate, a priori estimate of the exact amount of d.o.f. to parameterize the Earth model which is compatible with observed data and is not biased by artifacts resulting from the inversion scheme. The classical Backus-Gilbert (B-G) formulation of linear geophysical inverse problem [*Backus and Gilbert*, 1968; *Backus*, 1970a, 1970b; *Gilbert*, 1971] treats the model variation as continuous spatial and/or temporal function directly and interprets the inverse model as the actual Earth structure viewed through an imperfect window, the resolution kernel, which is constructed from weaving the available data kernels. Whereas conventional models are usually constructed by considering just the data fitting (in addition to a certain regularization scheme to deal with the intrinsic non-uniqueness embedded in almost all geophysical inverse problems), the resolution kernel revealed from the B-G formulation relates the estimated model to the “true” structure of the Earth. The appraisal of the resulting inverse model is made directly in the model space through the resolution kernel, rather than the conventionally indirect assessment in the data space by circuitously examining the data fitting. Because implementation of the B-G formulation requires the inversion of the Gram matrix formed by inner products among data kernels, it might be numerically impractical. This is due to the fact that Gram matrices of modern inverse problems constructed from data constraints are usually sizable and full, rather than sparse. Secondly, direct evaluations of inner products for certain forward approximations are sometimes inaccessible. For example, it is not possible to evaluate the integration of two crossing seismic rays for tomography based on ray theoretical formulation. The usual practice is to discretize the forward rule by choosing a certain ad hoc finite set of basis functions. After the finite parameterization, a resolution matrix analogous to the continuous resolution kernel is equally important in aiding the interpretation of the inverse model. The column vectors of the resolution matrix correspond to point spread functions (PSFs) [*Parker*, 1994; *Oldenborger and Routh*, 2009] which reveal how each impulse response of a specific model parameter is spread into the resolved inverse model estimates. The row vectors, on the other hand, associate with the averaging functions (AFs) [*Backus and Gilbert*, 1968] and indicate how a specific estimate of a model parameter is in fact composed by the weighted average of all “nearby” model parameters. The important information is accessible only by examining the complete resolution matrix, rather than the popularly improvised checkerboard test. For an example, the popular damped least squares (DLS) solution [e.g., *Lawson and Hanson*, 1974] is regularized by the minimum model norm criteria. It can be shown by examining the AF that the DLS solution is a realization through the Dirichlet kernel that bears significant ringing positive and negative sidelobes arising from attempting a finite approximation to an anticipated spatially localized window with broadband spectrum. This is a general aliasing problem. Imperfect model resolution is attributed mainly to the band limitation of the model information available from the finite amount, and usually, highly nonuniformly distributed data constraints.

[3] An additional complication that further biases the inverse model is the inevitable effect of magnifying the noise contamination embedded within the data. This usually results from those not well-constrained, usually high wave number, components invoked to push for highly localized spatial resolution. This is revealed by the model covariance matrix and is due to the fact that the content of uncorrelated data noise is inevitably manifested in high wave number model components. In other words, the fundamental reason for the intrinsic tradeoff of resolution versus variance for inverse models is embedded in the inevitable tradeoff of spatial versus spectral localizations of resolution kernels.

[4] The most fundamental mathematical concern of a finite approximation to a continuous function is the truncation error. For a chosen basis, say *β*_{i}(**r**), *i* = 1, …, *M*, expansion of the continuous model variation, *m*(**r**), such that *m*(**r**) − *a _{i}β_{i}*(

**r**)∥ ≤ ɛ, where

*a*

_{i}is the set of finite coefficients, and ɛ is the acceptable tolerance which converges to zero as

*M*increases toward infinity. In a continuous inverse problem, since we will never be able to know exactly the “truth,”

*m*(

**r**), it is thus difficult to validate whether our finite approximation, both the selected basis and the truncation level,

*β*

_{i}(

**r**),

*i*= 1, …,

*M*, are in fact appropriate.

*Trampert and Snieder*[1996] points out that the arbitrary truncation of high degree spherical harmonics (SpH) invoked in the parameterization of global inverse problem leads to

*spectral leakage.*It is worth pointing out here that whereas model aliasing is caused by the lack of adequate sampling of the model structure provided by the given data constraints, spectral leakage arises from inadequate truncation of the fine scale basis functions.

*Chiao and Kuo*[2001] derive the formal relation of the actual resolution operator of the truncated model with respect to the original continuous Earth structure. To compromise between the intrinsic tradeoff of the spatial versus spectral localizations, they also devise the multiresolution wavelet parameterization [

*Chiao and Kuo*, 2001;

*Chiao and Liang*, 2003;

*Chiao et al.*, 2006;

*Gung et al.*, 2009]. Although it is difficult to justify the appropriateness of a particular choice of finite parameterization scheme, it is straightforward to convert inverse models among different parameterization domains along with conversions of the important model resolution and covariance matrices. These conversions are capable of revealing the potential inconsistency that may arise from a particularly invoked parameterization or regularization scheme.